Deformations and q-convolutions. Old and new results

Abstract

This paper is the survey of some of our results related to $q$-deformations of the Fock spaces and related to $q$-convolutions for probability measures on the real line $\mathbb{R}$. The main idea is done by the combinatorics of moments of the measures and related $q$-cumulants of different types. The main and interesting $q$-convolutions are related to classical continuous (discrete) $q$-Hermite polynomial. Among them are classical ($q=1$) convolutions, the case $q=0$, gives the free and Boolean relations, and the new class of $q$-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of $q$-convolutions. The main result is the construction of Brownian motion related to $q$-Discrete Hermite polynomial of type I.